Do nine sober men and one drunk man constitute a minyan? While I know that one should not be davening if they are drunk (tipsy as well?) but does that negate the fact that there are ten (or more) men in the room with regard to having a minyan?

On a related note, should a person who is drunk/tipsy be saying amen, and if they do does it count?

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It is acceptable to proceed with chazarat hashatz, etc. if one of the 10 men making the minyan is still davening the silent shmoneh esrei, or even if one is asleep and cannot be woken, as long as there are still 6 men paying attention. This may not be a completely analogous case, and I could not find a definitive answer to your question, but I would guess that it would be ok to proceed with the minyan.

As in the examples given above, it would obviously better to have 10 sober men, just like it would be better to wait for the 10th man to finish davening or to have 10 awake men.

As a disclaimer: this is just my own reasoning and not definitive! Obviously CYLOR. It is definitely possible that this case is not analogous since someone who is drunk shouldn't be davening to begin with.